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Discrete p-Lag FIR Smoothing of Polynomial State-Space Models

With Applications to Clock Errors

LUIS MORALES-MENDOZA

Veracruz University

Electronics DepartmentVeracruz, 93390

MEXICO

[email protected]

OSCAR IBARRA-MANZANO, YURIY S. SHMALIY

Guanajuato UniversityDepartment of Electronics, DICIS

Ctra. Salamanca-Valle, 3.5+1.8km, Palo Blanco, Salamanca

MEXICO

[email protected], [email protected]

Abstract: A smoothing finite impulse response (FIR) filter is addressed for discrete time-invariant state-spacepolynomial models commonly used to represent signals over finite data. A general gain is derived for the relevant

p-lag unbiased smoothing FIR filter. Applications are given for the time interval errors of a local crystal clock

and the United States Naval Observatory Master Clock. An excellent performance of the best linear unbiased fit

is demonstrated along with its ability to extrapolate linearly the future clock behaviors, as required by the IEEE

Standards, that can be used for holdover in digital communications networks.

KeyWords: smoothing FIR filter, unbiased filter, polynomial model, clock time interval error

1 Introduction

Polynomial models are often used to represent signals

and images over a finite number of discrete points.Denoising of such models measured in the presence

of noise is efficiently provided using the finite im-

pulse response (FIR) p-lag smoothers [13], relatingthe estimate to the past discrete pointn +p,p


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in continuous time. Most generally, we thus have anexpansion [9]

xkn =Kkq=0

x(q+k)(nN+1p)q(N 1 +p)q

q! (2)

such that k= 1 gives us (1) and xKn = xK(nN+1p)holds true fork = K.

If we now suppose that x1n is measured as sn in

the presence of noisevn having zero mean,E{vn}=0, and arbitrary distribution and covariance Q(i, j) =E{vivj}for all i and j , then the signal and measure-ment can be represented in state space, using (2), with

the state and observation equations as, respectively,

xn = AN1+pxnN+1p, (3)

sn = Cxn+vn, (4)

where theK1state vector is given by

xn = [x1nx2n . . . xKn ]T . (5)

TheK Ktriangular matrix A is specified as

A =

1 2

2 . . . K1

(K1)!

0 1 . . . K2

(K2)!

0 0 1 . . . K3

(K3)!...

......

. . . ...

0 0 0 . . . 1

, (6)

the power Ai projects the state fromn N+ 1 pton, and the1Kmeasurement matrix is

C =

1 0 . . . 0

. (7)

3 Unbiased Smoothing FIR Filter

Smoothing FIR filtering ofx1n can be provided if to

represent (3) and (4) on an interval ofNpoints from

n N + 1 p to n+ p, p < 0, using recursively

computed forward-in-time solutions [16] as

XN(p) = ANxnN+1p, (8)

SN(p) = CNxnN+1p+ UN(p) , (9)

where

XN(p) =xTnp x

Tn1p . . . x

TnN+1p

T, (10)

SN(p) = [snpsn1p . . . snN+1p]T

, (11)

UN(p) =

vnpvn1p . . . vnN+1pT

, (12)

AN =

(AN1+p)T . . . (A1+p)T (Ap)TT

,(13)

CN=

CAN1+p

...

CA1+p

CAp

=

(AN1+p)1...

(A1+p)1(Ap)

1

, (14)

where(Z)1means the first row of a matrix Z.Given (8) and (9), the smoothing FIR filtering es-

timate ofx1n can be obtained as follows:

x1n|np =

N1+pi=p

hli(p)sni (15a)

= WTl (p)SN (15b)

= WTl (p)[CNxnN+1p+ UN(p)] ,

(15c)

where hli(p) hli(N, p) is the l-degree FIR filtergain [15] dependent on N and p [9, 10] and the l-

degree and1Nfilter gain matrix is given by

WTl (p) =

hlp(p) hl(1+p)(p) . . . hl(N1+p)(p)

(16)

that is to be specified in the minimum bias sense.

3.1 Unbiased Polynomial Gain

The unbiased smoothing FIR filtering estimate can be

found if we start with the unbiasedness condition

E{x1n|np}= E{x1n} . (17)

Combining E{x1n} = (AN1+p)1xnN+1p

with the mean estimate E{x1n|np} =

WTl (p)CNxnN+1p taken from (15c) leads tothe unbiasedness constraint

(AN1+p)1= WTl (p)CN, (18)

where Wl(p) means the l-degree unbiased gain ma-trix.

It has been shown in [9] that (18) can alternatively

be represented as

WTl (p)V(p) = JT , (19)

where J =

1 0 . . . 0 N

Tand the p-dependent and

N (l+ 1) Vandermonde matrix [17] is specified by

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V(p) =

1 p p2

1 1 +p (1 +p)2

1 2 +p (2 +p)2

......

...

1 N1 +p (N 1 +p)2

.

(20)

Because the inverse ofVT(p)V(p)exists, a solu-tion to (19) can be written as

WTl (p) = JT[VT(p)V(p)]1VT(p) . (21)

Following [18], the component of (16) can furtherbe represented with the degree polynomial

hli(p) =l

j=0

ajl(p)ij , (22)

where l [1, K], i [p, N1 +p], and ajl(p) ajl(N, p)are still unknown coefficients.

Substituting (22) to (19) and rearranging the

terms lead to a set of linear equations, having a com-

pact matrix form of

J = D(p)(p) , (23)

where J =

1 0 . . . 0 K

T,

=

a0(K1) a1(K1) . . . a(K1)(K1)

K T

, (24)

and a low dimensional,ll, symmetric matrix D(p)is specified via (20) as

D(p) = VT(p)V(p)

=

d0(p) d1(p) . . . dl(p)d1(p) d2(p) . . . dl+1(p)

......

. . . ...

dl(p) dl+1(p) . . . d2l(p)

.(25)

The component in (25) can be developed as in

[10],

dm(p) =

N1+pi=p

im , m= 0, 1, . . . 2l , (26)

= 1

m+ 1[Bm+1(N+ p)Bm+1(p)] ,

(27)

whereBn(x)is the Bernoulli polynomial.

An analytic solution to (23) gives us the coeffi-cient

ajl(p) = (1)j

M(j+1)1(p)

|D(p)| , (28)

where|D| is the determinant ofD(p), turning out tobep-invariant, andM(j+1)1(p)is the minor ofD(p).

3.2 p-Lag Polynomial Ramp Unbiased Gain

It can now be shown that the 1-degreep-shift polyno-mial ramp unbiased gain existing fromptoN 1 +pis specified, by (22) and (28), as

h1i(p) =a01(p) +a11(p)i , (29)

with the coefficients

a01(p) =2(2N1)(N1) + 12p(N1 +p)

N(N2 1) ,

(30)

a11(p) =6(N1 + 2p)N(N2 1)

. (31)

In turn, the noise power gain (NPG) associated

with (29) is provided to be [4]

g1(p) = a10(p)

= 2(2N 1)(N 1) + 12p(N 1 +p)

N(N2 1) .

(32)

We notice that, by p = 0, the gain (29) becomes

that originally derived in [19] via linear regression andrederived in [15] in state space. This gain was further

modified to be optimal in the minimum MSE sense

in [20] and, in [4], one can find the higher degree gains

along with the relevant NPG analysis.

4 Smoothing of Clock Errors

In this section, we apply the smoothing FIR filtering

algorithm to provide smoothing of clock error models

measured in the presence of noise.

4.1 Two-state crystal clock model

In the first experiment, the time interval error (TIE) of

a crystal clock imbedded in the Stanford FrequencyCounter SR620 was measured each second during

357332 s using another SR620 for the reference ce-

sium clock (Symmetricom CsIII) as shown in Fig. 1

as xn+ noise. On the interval of N = 357333points, the clock was identified to have two states. For


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357332+px

0 1 2 3

5

10

Time, s105

nx

35.9357332

n

x clock noise

x

p = 0

p = 3 57 33 2

85.00 x

Figure 1: Unbiased FIR filtering and smoothing of the

crystal clock TIE.

the two-state model, the ramp FIR smoother was or-

ganized by changing a variable in (15a) and (29) to

produce the estimate at n+pas

x1(n+p)|n =N1i=0

h1i(N, p)sni, (33)

where

h1i(N, p) =2(2N 1)6i

N(N+ 1) +

6p(N 12i)

N(N2 1) .

(34)

To find the best unbiased fit, all the data must beinvolved. We thus substituteNwithn +1and modify(33) to

xn+p=ni=0

h1i(n+ 1, p)sni, (35)

where

h1i(n+ 1, p) =2n(2n+ 1)6in+ 6p(n2i)

n(n+ 1)(n+ 2) .

(36)

The best linear unbiased fit appears if to smooththe data at the initial point with p = n as x0 andfilter at the current point withp = 0as xn. A straight

line xn passing through two these points is providedwith

xn+p = x0+ (n+p)xn x0

n (37a)

=ni=0

h1i(n+ 1, p)sni (37b)

if we fix n and change p fromn to 0, as a variableforh1i(n+ 1, p)given with (36).

An evolution of the last point of the best fit pro-vided by the filtering estimate with p = 0at n is rep-resented in Fig. 1 with xn. This estimate is obtainedby (35) with n changed from 0 to n = 357332. Ascan be observed,xnunbiasedly tracks the mean of thexn+clock noise.

The best linear unbiased fit is shown in Fig. 1

for all the measured points as a straight line x357332+p

having the values ofx0= 0.85ns withp = 357332and x357332 = 9.35 ns with p = 0. Similarly, it anbe provided employing the higher degree gains. In

each of the cases, the fit will hold true only for the

observed database. It would be corrected for every

new measurement point added to the data.

4.2 Two-state Master Clock model

In the second experiment, we smoothed errors in the

USNO Master Clock. The USNO has published on

the WEB site the UTC UTC(USNO MC) time dif-ferences (73 points) measured each 5 days in 2008, as

issued monthly by BIPM [units are in Modified Julian

Dates (MJDs) and nanoseconds]. For this measure-

ment, we form the time scale starting with n = 0(54464.0 MJD) and finishing at n = 72 (54829.0MJD). The measurement is shown in Fig. 2a (circles).

Because the frequency drift in the USNO MC is

negligibly small and measurement is provided withnegligible errors, the clock was represented with two

states,K= 2,

xn = Axn1+ wn, (38)

sn = Cxn, (39)

in which

xn=

xnyn

, A =

1 0 1

, C = [ 1 0 ] .

The time scale of the USNO MC is corrected.Therefore, the time error noise can be considered to

be zero-mean and white. We calculate the variance

of this noise as 2xn = E{w2n} providing the aver-

aging from zero to the current point n. The secondclock state can be represented by the time derivative

of the first state and the noise variance calculated sim-ilarly. Accordingly, we form the clock noise vector as

wn= [ wxn wyn]T and the covariance matrix with

n =

2xn E{wxnwym}

E{wynwxm} 2yn

,

where0 m n. Note that the effect of frequencynoise is relatively small in n.


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010 20 30 40 50 60 70

5

5

Time, 5 days

Optimal

UnbiasedUSNO MC

(a)

010 20 30 40 50 60 70

5

5

Time, 5 days

Optimalsmoothing

Unbiased smoothing

USNO MC

Unbiasedprediction Optimalprediction

(b)

010 20 30 40 50 60 70 80

2

4

6

Time, 5 days

Unbiased

Optimal

(c)

Figure 2: FIR estimates of the USNO MC current

state via the TIE measured during 2008 each 5 days:

(a) filtering of the first state xn, (b) prediction and

smoothing of the first state, and (c) filtering of the sec-

ond stateyn.

For (38) and (39), the optimal FIR filtering esti-mate can be provided, with p = 0, as follows. GivenSn,Cn,A, andZ, the optimal FIR filtering estimateis obtained forn K= 2by solving the DARE [21]

0 = Z0Z1 Z0+ 2Z0+

Z SnSTn

Z1 Z0, (40)

for Z0 and then computing

xn|n= An(CTnCn)

1CTnZ0(Z0+Z)1Sn. (41)

By neglecting Z in (41), one can also providethe unbiased FIR filtering estimate

xn|n= An(CTnCn)

1CTnSn. (42)

Figure 2 sketches the optimal and unbiased FIR

estimates of the clock first state xn and second state

yn. Figure 2a reveals thatxn and xn differ in averageon about 15%and that this measure can reach 50%at

some points. Although a comparison of optimal andunbiased estimates is a special topic, there is an im-

mediate explanation to differences with small N. The

unbiased filter provides the best fit for the noisy pro-

cess, whereas in the optimal filter this fit is adjusted

by the noise covariance function. Note that the lattercannot be ascertained correctly with a small number

of measurements. Therefore, the optimal filter may

produce errors. On the other hand, the unbiased fil-ter is associated with large horizons and may not beprecise otherwise. So, we have an inconsistency that

would be reduces by increasing N.

Smoothing of time errors is illustrated in Fig. 1b.

Here, we employ thep-shift general optimal algorithm[21] the mean square initial state function Z0 deter-

mined by (38) and the p-lag FIR smoothing estimate

(35) with (34). For the illustrative purposes, we fix

three time points, n = 21, n = 36, andn = 56, andallowp < 0for smoothing and p > 0 for prediction.These estimates are depicted in Fig. 2b with the right

and left arrows, respectively. It can be shown that the

unbiased prediction is exactly that found in [22] em-

ploying the ramp FIR filter. In [22], one can also findan unbiased prediction of clock errors obtained with a

5-point step and unbiased smoothing of measurement

providing the best fit. An important observation can

be made observing Fig. 1b. It is seen that the unbiasedsmoothing function ranging fromn = 36 to zero fitsbetter than the optimal one. However, this fact does

not mean that the optimal smooth is lesser accurate,

because the latter fits better the full measurement.

Finally, Fig. 1c sketches estimates of the second

clock state provided with the optimal and unbiased fil-

ters. Again we infer that both estimates are consistent,except for the region from n= 25to n= 45, in whichthe discrepancy reaches50% (Fig. 2b).

5 Conclusions

In this paper, we discussed the unbiased smoothingFIR filtering of discrete-time polynomial signals rep-

resented in state space. The relevant gain for the FIR

structure was found in the matrix form with no re-

quirements for noise and initial conditions. This gain

was also developed in the unique polynomial formhaving strong engineering features. As examples of

applications, we provided smoothing of time errors in

a crystal clock and in the USNO Master Clock. A high

efficiency of the proposed solution is demonstratedgraphically. We finally notice that, even visually, the

best linear unbiased fit shown in Fig. 1 provides us

also with the best extrapolation (prediction) of future

clock behaviors that is required by the IEEE Standard

1139 [6]. This fit can be used for holdover in steer-


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ing of local clocks via the Global Positioning System(GPS) timing signals in digital communications and

other networks, for example.

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